Thanks to Joe, I've discovered that it is indeed a gear that is the culprit in my grandmother's Singer 413, and I've ordered the part from partsonline dot com. Also ordered a new timing belt just in case it needs replacing. I haven't examined it that closely, however after all these years of not working, it probably has dry rot.
Called the repair center today and they told me I needed to bring it in for an estimate on replacing the gear. I did find a tutorial on how to DIY, but I'm no expert in repairing machines, so I'll see what estimate the local man comes up with.
The attachments my mom gave me with the machine don't fit that machine. They fit her old White Domestic machine that she sold some years ago for the paltry sum of $10. She remembers giving the buyer some attachments, which probably were the ones that fit the 413. I told her no problem, because I have other Singer attachments I can use if needed.
Anyone need attachments for a White Domestic?